Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
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Is there a closed form solution?
Question:
Let $A\in\mathrm{GL}(n,\mathbb{R})[[u]]$ with $A=A^T$, that is a symmetric real matrix, which can not be decomposed into a blockmatrix with $0$ blocks, such that
$$ A = \begin{pmatrix} A_1 &...
- 121
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Degrees of freedom of an $r$-ranked tensor?
I'm trying to determine the degrees of freedom for parameters of a tensor in shape $(J_1,\dots,J_D)$ and with rank $r$, where "rank" refers to the smallest number of rank-1 tensors whose sum ...
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Inverse and Determinant of Matrix $Axx^TA+cA$
Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
- 2,497
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If I can solve $Ax = b$ efficiently, is there a method to solve $(I+A)x=b$ efficiently?
Say if I already have the LU factorization of a square matrix $A$, is there an efficient way to get the LU factorization of $I+A$? (We may assume all the matrix I mentioned is invertible.)
I know from ...
- 555
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Best method for sequential small size Hermitian smallest eigenpair problem
I am working on a perhaps rather strange problem. To find the smallest eigenvalue and its eigenvector, for a large number (a few billions) of small (20 * 20 to 200 * 200) Hermitian matrices. These ...
- 31
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If $A^*A=A^*B=B^*A=B^*B$, prove that $A=B$.
Suppose that $H$ is a Hilbert space and $A,B\in B(H)$ are such that $A^*A=A^*B=B^*A=B^*B$,then $A=B$.I'm having difficulty trying to read a proof that solves this problem.
First, the author decomposes ...
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Fast way to compute the largest eigenvector of an expensive-to-compute matrix
Consider an $N \times N$ Hermitian positive semidefinite matrix $M$. Computing the elements of $M$ is expensive so we wish to compute as few as possible. We can assume that $M$ can be approximated as ...
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$O$ orthogonal with $\det(O)=-1$ implies $||\Omega - O \Omega O^{T}|| = 2 $?
Let $O\in \mathrm{O}(2n)$ be an orthogonal matrix. Let $\Omega$ be the matrix $\Omega:=
\bigoplus^n_{i=1} \begin{pmatrix}
0 & 1 \\
-1 & 0 \\
\end{pmatrix}.$
Is it true that:
$\det(O)=-1$ ...
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Quadratic form of a matrix where non-standard decomposition is known
Let $1\le m<<n$ for integers $m,n$. I have two symmetric matrices of size $n\times n$, say $A$ and $B$. My goal here is to know a simple form, or an upper bound on the following quadratic form:
$...
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Minimize $||A-AWW^TA^T||_F$ w.r.t. $W$
Given $n \in \mathbb{N}$ and $A \in \{0,1\}^{n \times n}$, we aim to find
$$\arg \min_{W \in \mathbb{R}^{n \times n}} f(W) = ||A-AWW^TA^T||_F,$$
where $||\cdot||_F$ represents the Frobenius norm with
$...
- 2,150
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Given a singular matrix $B$ and a result $C=A\times B$, find matrix $A$ over finite fields.
The problem is a conituation of this problem, but over finite fields.
In the context of a finite integer field, particularly when all entries in matrices $A, B$, and $C$ are drawn from a finite ...
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Euclidean norm of a vector resulting from a matrix multiplied by an orthonormal matrix multiplied by an arbitrary vector!
Let $A \in \{0,1\}^{m\times n}$ be a binary matrix with $ m < n$, and let $U \in \mathbb{C}^{n \times n}$ be an arbitrary orthonormal matrix. Let $\sigma_{min}$ denote the smallest non-zero ...
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Binary matrix power for a specific entry.
Consider a square $n\times n$ matrix $A$ whose entries are binary, that is, for all $i, j\in [n]$, it holds that $A_{i, j} \in \{ 0, 1\}$.
I am interested in the following decision procedure:
Given a ...
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Understanding QR decomposition in the context of a cubature Kalman filter
I am working on implementing a square-root cubature kalman filter based on the algorithm presented in this conference paper, as well as this paper more broadly.
I have got the algorithm to run; ...
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Finding Correspondence between Matrices after Decomposing a Hermitian Matrix
After decomposing the Hermitian matrix $M$, I obtain a set of matrices $D_j$, where each $D_j$ is defined as
$D_j = \lambda_j u_j u_j^H$,
$\lambda_j$ and $u_j$ are eigenvalues and eigenvectors, and $M$...
